Economies of Scale

Rural, Urban, and Regional Economics ECON 376 (by Prof. Kilkenny)

An economy of scale is a decrease in the average cost associated with an increase in the quantity. Increasing returns to scale is when profits are higher for larger quantities. This note describes how to identify, measure, and define the relevant concepts.

I. The total costs of production (“TC”) at an establishment include sunk or fixed costs (“K”) and costs that vary with the level of output, called variable costs, written v(Q):

1. TC = K + v(Q)

where K includes the cost of the site and all other things that must be paid for even if the firm produces or sells nothing. For retail businesses, labor may be a fixed cost because someone must be present (and paid) at all times when the shop is open even if nothing is sold.

The simplest form of variable costs are constant per unit output:

v(Q) = v∙Q. In this case, TC = K + v∙Q.

II. Average cost (AC) is the total cost per unit quantity produced:

2. AC = TC/Q = K/Q + v , in this case. See that AC declines as Q increases by noting that the first derivative of AC with respect to Q = -K/Q² is negative.

III. Marginal cost (MC) is the additional cost of producing an additional unit:

3. MC = ∂TC/∂Q = v in this case.

Graphically, it is easy to see that whenever there are fixed costs, average cost always declines as more is produced. Also, for this example where TC=K +vQ, marginal costs are constant over all levels of output Q:

IV. Variable costs v(Q) include the costs of raw materials and other industry services, labor, energy, taxes, and transport:

4. v(Q) = Σ_i Q∙q_i·P_i + w_r·L∙Q + P_rE·q_E·Q + taxes + transport

where q_i denotes the quantity of input from industry "i" required per unit of output, so that Q∙q_i measures the amount of input i used; and Q∙q_i·P_i is the cost of that input. Sum over all inputs to find total variable input costs. Again, remember that for service sector businesses, at least some of the labor is a fixed cost because someone must be present (and paid) at all times when the shop is open, even if nothing is sold.

w_r is the regional wage rate, L is the labor per unit output (invert this for the measure of productivity: output per unit labor), so w_r·L∙Q is labor costs (for variable labor inputs). Finally, q_E denotes the quantity of energy required per unit of output, so that Q∙q_E·P_rE is the cost of energy used by the establishment in location r. Taxes and transport costs will be detailed later.

V. Divide through by Q to show that for our example, variable costs are constant per unit output:

5. v = Σ_i q_i·P_i + w_r·L + P_rE·q_E + tax per unit + transport per unit,

which is why MC is flat in the graph above, and why v(Q) = v∙Q, in this case.

VI. Profit is the surplus of revenues over costs:

Profit = R - TC = P∙Q - K - v∙Q = (P-AC)Q

6. Profit = P∙Q – K - Σ_i Q∙q_i·P_i + w_r·L∙Q + P_rE·q_E·Q + taxes + transport

VII. Increasing returns to scale is when profits per unit rise as quantity rises. Profits per unit are:

7. Profit/unit = P – K/Q - Σ_i q_i·P_i + w_r·L + P_rE·q_E + tax per unit + transport per unit

Profit/Q = P- K/Q – v

Formula 7 highlights the three main sources of increasing returns: higher prices paid, more output per plant, and lower marginal costs.

When v falls with Q it is called a technological source of internal increasing returns if it is due to reductions in q_i (input productivity), L , (labor productivity), and/or q_E (energy productivity).

Increases in output price (P) or reductions in input prices (P_i or P_E), or wages (w), are called pecuniary sources of increasing returns to scale. But those elements are not under the control of an individual firm. They are not, therefore, sources of increasing returns to scale that are internal to an individual firm. They are usually affected, however, by the numbers of firms in a location, and thus these are often the sources of external returns to scale known as agglomeration economies of scale.

VIII. Economies of scale can be classified a dozen ways:

Type of economy of scale:				affected element (formula 7), or type (this list)
internal	1.pecuniary			P, K
	technological		2. static technological	q, L
			3. dynamic technological	q, L
external	localization	static	4. “shopping”	P, Q
			5. "Adam Smith" specialization	1,Pi
			6. "Marshall" labor pooling	2
		dynamic	7. "Arrow" learning by doing	3, L
	urbanization	static	8. "Jane Jacobs" innovation	1,2,3
			9. "Marshall" labor pooling	2,w
			10. "Adam Smith" division of labor	1,2,Pi
		dynamic	11. "Romer" endogenous growth	3,P
			12. pure agglomeration	transport, taxes

IX. External or agglomeration economies (#4-#12, above) arise from an increase in the economic activity in the location which cause, through 8 various mechanisms, industry-wide costs to fall or revenues to rise. Localization economies (#4-7), arise when there are large numbers of firms in that same industry and same place. Urbanization economies (#8-#11) arise from the presence of a large number of different industries in the same place. Static economies of scale arise contemporaneously. Dynamic economies of scale are reductions in cost or increases in revenue per unit that arise from repeated and continuous production activity over time (“practice makes perfect”).

#4: Shopping Goods static Localization economy of scale:

Firms that supply shopping goods (where customers do the transport) enjoy more sales and higher revenues by locating close together. Shoppers are more attracted to the sites where there are many shops than few, because the act of shopping entails the fixed cost of driving to a store. The shoppers' choices to minimize transport costs per shop give rise to benefits to the stores that locate where lots of other stores are. The more shops there are in the same place, the higher are sales (Q), prices (P_o), and the higher are revenues.

All other localization economies arise from industry-wide cost reductions.

#5: Adam Smith static Specialization localization economies of scale:

Adam Smith wrote, "The division of labor is limited by the extent of the market." As we already know, the existence of fixed costs gives rise to internal economies of scale. We have also shown (homework) that the number of businesses a place can support is limited by the number of customers within the business’s spatial demand cone. Some places can support only one establishment, others aren’t big enough even for one. The more customers in a location, the more firms there can be in that location, and the more specialized each firm can be.

When demand is low, only small scale production (low K) is feasible. It is not feasible to proliferate fixed costs (have lots of Ks) so that each establishment can specialize. So in small markets, businesses have to take care of most operations in-house. All parts, for example, may be fabricated within one plant when that plant’s market is small. They cannot afford to out-source. That’s what Adam Smith meant when he said the extent of the market limits the division of labor.

As the market expands, a preexisting plant (in the X industry) can enjoy profits due to scale economies. The profitability of industry X entices new firms in the SAME industry to open. As the number of firms rise, the local demand for inputs for the X industry also rises. At some point, the market for inputs for X becomes large enough to support plants that specialize in making those inputs (subsets of the previously integrated process X):

raw materials processing

for inputs to X à inputs for X à industry X à buyers of X

“upstream” (relative to industry X) àààà “downstream” from X

This outsourcing allows both the upstream input supplying plants and downstream industry plants to profit from economies of scale and the productivity gains due to specialization ( q↓ and L↓). This makes the costs of outsourced inputs (P_i) lower than the cost of producing everything internally-- when the market is large enough.

(Note: We don’t call land at the plant site an “input.” So: Don’t jump to the incorrect conclusion that increasing numbers of firms in a location always drives input prices up. (It drives rents up, but because we do not call land an “input,” this effect is not the point.) In fact, localization economies of scale is the self-explanatory name given to the observation (facts) that input prices are often lower where there are many firms in the same industry.)

#10. "Adam Smith" static urbanization economies of scale

are similar to the localization version (#5 above), the main difference being that the division of labor upstream is be made possible by the existence of many different downstream buying industries in the same place.

#6. Marshallian Labor-Pooling static localization economies of scale:

The famous twentieth century economist Alfred Marshall also observed that labor costs for localized firms were also lower—exactly the opposite of our naïve expectation that increased numbers of businesses in a place would bid wages up. While wages may indeed be bid up by competition for workers, overall labor costs, which include screening applicants, training, benefit packages, etc, are likely to be lower where there are many firms in the same industry. Workers with the industry-specific skills are likely to be attracted to the location, the labor market becomes “deep,” and both recruiting expenses and training costs are lower for the local firms.

Workers are also LESS nervous about being laid off for no good reason from one firm in a location where there are other firms at which the worker’s industry-specific knowledge would be valued. That makes severance costs lower. Plus, firms don’t have to pay premium wages to compensate their employees for a dearth of local opportunities for career advancement. Thus, businesses can more easily expand (or contract) in “deep” labor markets.

In deep labor markets, however, workers are MORE aware that if they shirk or are unproductive, they can be easily replaced. So, productivity is maintained at a higher level in deep labor markets (q↓ and L↓ == fewer inputs and less labor needed per unit output). Thus, labor costs of all kinds are LOWER when there are lots of firms in the same industry in the same location, and firms there enjoy external localization economies of (industry-wide) scale.

#9. Marshallian labor pooling static urbanization economies of scale : are similar to the localization version (#6 above), but the benefit arises from the breadth or diversity of types of industries in the same location. Workers are attracted to places with different opportunities because, for example, they want to avoid being stuck in a rut. Firms are attracted because workers with skills in one industry may bring productive innovations to firms in other industries.

Using Mass production techniques, businesses capture economies of scale. Using mass customization techniques, businesses capture economies of scope. Urbanization economies of scale are similar to economies of scope.

#7. Dynamic Arrow “learning-by-Doing” localization economies of scale: are reductions in costs or increases in revenue per unit that arise from repeated and continuous production activity over time in the same place (“practice makes perfect”). Productivity improvements can reduce input waste (q↓) or reduce labor requirements (L↓) so that costs are lower. Or, costless quality improvements allow a business to obtain higher prices for their product without losing customers.

The places where an industry initially achieves critical mass (Adam Smith static localization economy of scale) are the places where firms in that industry most likely to enjoy the dynamic “Learning-by-Doing” economies of scale. That is another reason why places tend to specialize in certain industries over time.

Dynamic localization economies are observed as follows: Measure the size of an industry in a location at time t by employment: E_irt . Time series evidence of dynamic localization economies of scale is shown if E_irt is statistically significantly positively related to prior period (t=0) E_ir0.

#8. "Jane Jacobs" innovation urbanization economies of scale arise because there is more innovative activity in places with more variety. Remember that innovation is the application of an existing invention to a different purpose. The more different things are being done locally, the opportunity there is for observing and adapting technologies or ideas from one sector to other sectors. Thus, higher industrial diversity supports more innovation and higher efficiency, so that firms operate at lower costs, and higher profit.

#11. "Romer" endogenous growth

When there are agglomeration economies of scale, average and marginal costs are lower the larger is a location. Thus, the larger the market, the more profit can be earned, the more attractive the location to firms, the more jobs there are in the place, the more labor pools there, the larger the market, and so on. This positive feedback from the size of an industry in a location drives the spatial concentration of economic activity and the growth of cities.

#12. pure agglomeration economies of scale arise from spreading the fixed costs of city infrastructure (roads, utilities, civic administration, etc) over more taxpaying businesses and residents. Thus, bigger cities are cheaper per citizen to live in that smaller cities. Note that on the other end of the place-size scale, there are diseconomies of agglomeration, such as the discomforts of overcrowding, heavy traffic, pollution, crime, etc.